Instructions

For the pasta sheets: Process spinach and eggs in food processor until spinach is finely chopped, about 30 seconds, scraping down sides of bowl as needed. Add flour and process until mixture forms cohesive dough that feels soft and is barely tacky to touch, about 45 seconds. (If dough sticks to fingers, add up to ¼ cup extra flour, 1 tablespoon at a time, until barely tacky.)
Transfer dough to clean counter and knead by hand to form smooth, round ball, about 2 minutes. Cover tightly with plastic wrap and let rest at room temperature for at least 15 minutes or up to 2 hours.
Transfer dough to clean counter, divide into 10 pieces (about 2¾ ounces each), and cover with plastic. Flatten 1 piece of dough into ½-inch-thick disk. Using pasta machine with rollers set to widest position, feed dough through rollers twice. Bring tapered ends of dough toward middle and press to seal. Feed dough seam side first through rollers again. Repeat feeding dough tapered end first through rollers set at widest position, without folding, until dough is smooth. (If dough sticks to fingers or rollers, lightly dust with flour and roll again.)
Narrow rollers to next setting and feed dough through rollers twice. Continue to progressively narrow rollers, feeding dough through each setting twice, until dough is thin but still sturdy; transfer to lightly floured counter. Using pizza cutter or sharp knife, trim pasta sheet into 11 by 3 ½-inch rectangle; discard scraps. Let lasagna noodle sit uncovered on counter while rolling remaining 9 pieces of dough (do not overlap noodles as they may stick together).
Bring 4 quarts water to boil in large pot. Add noodles and salt and cook, stirring often, until just tender, 2 to 3 minutes. Drain noodles and toss with oil. Using tongs, lay noodles flat over 2 rimmed baking sheets, overlapping as needed, and let cool slightly.
For the lasagna: Adjust oven rack to middle position and heat oven to 425 degrees. Grease 13 by 9-inch baking dish. Stir 3 cups Parmigiano, cottage cheese, cream, garlic, cornstarch, pepper, and salt together in bowl.
Spread 1 cup ragù in bottom of prepared dish. Lay 2 noodles lengthwise in dish, trimming edges as needed to fit. Spread 1 cup cream sauce over top, followed by second layer of noodles. Spread 1 cup ragù over top, followed by third layer of noodles. Repeat layering of cream sauce, noodles, and ragù, ending with fifth layer of noodles. Spread remaining cream sauce over top and sprinkle with remaining ¼ cup Parmigiano. Spray sheet of aluminum foil with vegetable oil spray and cover lasagna. (Lasagna can be refrigerated for up to 24 hours.)
Bake until bubbling around edges, about 30 minutes. Remove foil and continue to bake until top is spotty brown, about 10 minutes. Let lasagna cool for 45 minutes. Cut into pieces and serve.